70 Royal Society, 



ber considerably above ten thousand, with thermometers placed on 

 nearly a hundred different substances, exposed to the open air, under 

 different circumstances, and in various states of the sky, at the Royal 

 Observatory at Greenwich. 



Feb. 18. — " On the Diurnal Variation of the Magnetic Declina- 

 tion of St. Helena." By Lieut.-Colonel Edward Sabine, R.A., For. 

 Sec. R.S. 



It has long been known that the diurnal variation of the magnetic 

 needle is in an opposite direction in the southern, to what it is in 

 the northern hemisphere ; and it was therefore proposed as a pro- 

 blem by Arago, Humboldt and others, to determine whether there 

 exists any intermediate line oi' stations on the earth where those 

 diurnal variations disappear. The results recorded in the present 

 paper are founded on observations made at St. Helena during the 

 five consecutive years, from 1841 to 1845 inclusive; and also on 

 similar observations made at Singapore, in the years 1841 and 1842; 

 and show that at these stations, which are intermediate between the 

 northern and southern magnetic hemispheres, the diurnal variations 

 still take place ; but those peculiar to each hemisphere prevail at 

 opposite seasons of the year, apparently in accordance with the 

 position of the sun with relation to the earth's equator. 



Feb. 25. — " On certain Properties of Prime Numbers." By the 

 Right Hon. Sir Frederick Pollock, M.A., F.R.S., Lord Chief Baron 

 of the Exchequer, &c. 



The author of this paper, after noticing Wilson's Theorem, (pub- 

 lished by Waring about the year 1770, without any proof), which 

 theorem is that, if A be a prime number, 1. 2. 3. . . . (A — l)-f-l is 

 divisible by A ; refers to Lagrange's and Euler's demonstrations, 

 and mentions Gauss's extension of the theorem, to any number, not 

 prime ; provided that instead of 1, 2, 3, &c. (A — 1), those numbers 

 only be taken which are prime to A, and 1 be either added or sub- 

 tracted. This theorem was published by Gauss without a proof in 

 1801, with a rule as to the cases in which 1 is to be added or sub- 

 tracted, the correctness of which is questioned by the author, who 

 proceeds to propound the following theorem, which he had previ- 

 ously, for distinctness, divided into three. 



If any number, prime or not, be taken, and the numbers prime to 

 it, and less than one half of it be ascertained, and those be rejected 

 whose squares +1 are equal to the prime number, or some multiple 

 of it (which may be more than one), then the product of the re- 

 maining primes (if any), + 1 shall be divisible by the prime number. 



He gives as examples, 14, the primes to which, and less than one 

 half, are 1, 3, 5, and 1.3. 5=15; therefore 1.3.5 — 1 = 14; also 

 15, the primes to Avhich and less, are 1, 2, 4, 7; but 4x4 = 16 

 = 15 + 1 ; therefore 4 is to be rejected, and 1. 2. 7 + 1 = 15. The 

 author adds another theorem, that if A be a prime number, all the 

 odd numbers less than it (rejecting as before) ; also, all the even 

 numbers (making the same rejection except A — 1) will, multiplied 

 together, be equal to A+1. 



The author then proceeds to prove Gauss's extension of Wilson's 



